The Area Between
Name ___________________________
AreaBetween.tns
Class ___________________________
Problem 1 – Making Pathways
While integrals can be used to find the area under a curve, they can also be used to find
the area between curves through subtraction (just make sure the subtraction order is the
equation of the top curve minus the equation of the bottom curve.)
Suppose you are a building contractor and need to know how much concrete to order to
create a pathway that is 1/3 foot deep. The sidewalk borders can be modeled by
f(x) = sin(0.5x) + 3 and g(x) = sin(0.5x) from –2 to 2.
To find the volume, all that is needed is to multiply the area of the sidewalk by the depth
of the sidewalk.
On page 1.4, graph the functions and then calculate the
integrals of f1 and f2 (Menu > Analyze Graph >
Integral).

What is the value of the integral of f1? Of f2?
Now use the Text and Calculate tools to find the volume of the pathway.

What is the formula for the volume of the sidewalk?

How much concrete is needed for the pathway?
Verify your calculations on page 1.6. Use the Numerical Integral command (nInt) in the
Numerical Calculations menu to find the area.
Note: The nInt command has the following syntax nInt(expression, variable, lower bound,
upper bound). Hint: The expression is the difference of the functions.
©2012 Texas Instruments Incorporated
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The Area Between
The Area Between
Problem 2 – Finding New Pathways
The owners have changed the design of the pathway. It will now be modeled by
f(x) = x(x + 2.5)(x – 1.5) + 3 and g(x) = x(x + 2)(x – 2) from –2 to 2.
On page 2.2, graph the functions and then calculate the
integrals of f1 and f2.

What is the value of the integral of f1? Of f2?
Now use the Text and Calculate tools to find the
volume of the pathway.

How much concrete is needed for the pathway?
Verify your calculations on page 2.3.
Problem 3 – Stepping Stones
The owners also want stepping stones, which can be modeled by f(x) = – (x – 1)(x – 2) +
2 and g(x) = (x – 1)(x – 2) + 0.5. This situation different because the starting and
stopping points are not given. Assume that the stepping stones are 1/3 foot thick.
On page 3.2, graph the functions. Use the Intersection
Point(s) tool, to display the coordinates of the
intersection points of f1 and f2.

What are the coordinates of the two intersection
points?
Store the x-value of the left point as a and the x-value of
the right point as b. To do this, click on the value, press
/ + h, type the letter, and then press ·.
Note: All variables and values linked to variables appear in bold type.
Calculate the integrals of f1 and f2. Then use the Text and Calculate tools to find the volume
of the pathway.

What is the value of the integral of f1? Of f2?

How much concrete is needed for the pathway? Verify your calculations on page 3.3.
©2012 Texas Instruments Incorporated
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The Area Between